Unifying the Robertson-Walker metric

As Stephen Hawking "unified" the Schwarzschild metric, yielding Hawking radiation, could one modify the Robertson-Walker metric to satisfy most generally both the cosmological principle and quantum mechanics?

Do you mean quantum field theory in a Robertson-Walker background? This is indeed possible. Actually Hawking was one of the authorities in this subject in the 80's and he did one of the first calculations of the spectrum of perturbations (as the perturbations have their origin as quantum fluctuations in an de-Sitter expanding background). There is a very nice paper of Alan Guth in arxiv.org explaining the story about their meeting in Cambridge and how the first "predictions" of inflation came into existence (you can find it searching for hawking and birthday).